rna secondary structure -...

RNA Secondary Structure

CSE 417W.L. Ruzzo

The Double Helix

Los Alamos Science

The “Central Dogma” ofMolecular Biology

DNA RNA Protein

DNA (chromosome) RNA

(messenger)

Proteingene

cell

Non-coding RNA

• Messenger RNA - codes for proteins• Non-coding RNA - all the rest

– Before, say, mid 1990’s, 1-2 dozen known(critically important, but narrow roles: e.g.ribosomal and transfer RNA, splicing, SRP)

• Since mid 90’s dramatic discoveries• Regulation, transport, stability/degradation• E.g. “microRNA”: hundreds in humans• E.g. “riboswitches”: thousands in bacteria

DNA structure: dull

…ACCGCTAGATG…

…TGGCGATCTAC…

• RNA’s fold,and function

• Nature useswhat works

RNAStructure:Rich

Why is structure Important?

• For protein-coding, similarity in sequence is apowerful tool for finding related sequences– e.g. “hemoglobin” is easily recognized in all

vertebrates• For non-coding RNA, many different

sequences have the same structure, andstructure is most important for function.– So, using structure plus sequence, can find related

sequences at much greater evolutionary distances

Q: What’s so hard?

AC

U

G

C

A

G

G

G

A

G

C

AA G

C

GA

G

G CC

U

C

UGC A

A

UG

A C

G

GU

G

CA

U

GA

G

A G

C

G UCU UU

U

C

A

A

CA

C UG

U

UA

U

G

G

A

A

G U

UU

G

GC

UA

GC

G UU C

UA

G

AG

C U

G

UG

A

CA

C

UG

CC

G

C

GA

C

G

G GA

A

A

GU A A

C

GG

G

CGG

C

G

A

GU

AA

A

C C

C

GA

UC CC

G

GU

G

A

A

U

AG

CC

U GA

A

A

A

A

CA

A

A

GU

A

CA CGG

G

A

UAC

G

A: Structure often more important than sequence

6S mimics anopen promoter

Barrick et al. RNA 2005Trotochaud et al. NSMB 2005Willkomm et al. NAR 2005

E.coli

Chloroflexus aurantiacus

Geobacter metallireducensGeobacter sulphurreducens

Chloroflexiδ -Proteobacteria

Symbiobacterium thermophilum

Used by CMfinder

Found by scan

“Central Dogma”=

“Central Chicken & Egg”?DNA RNA Protein

Was there once an “RNA World”?

DNA (chromosome) RNA

(messenger)

Proteingene

cell

6.5 RNA Secondary Structure

Algorithms

RNA Secondary Structure

RNA. String B = b1b2…bn over alphabet { A, C, G, U }.

Secondary structure. RNA is single-stranded so it tends to loop backand form base pairs with itself. This structure is essential forunderstanding behavior of molecule.

G

U

C

A

GA

A

G

CG

A

UG

A

U

U

A

G

A

C A

A

C

U

G

A

G

U

C

A

U

C

G

G

G

C

C

G

Ex: GUCGAUUGAGCGAAUGUAACAACGUGGCUACGGCGAGA

complementary base pairs: A-U, C-G

RNA Secondary Structure

Secondary structure. A set of pairs S = { (bi, bj) } that satisfy: [Watson-Crick.]

– S is a matching and– each pair in S is a Watson-Crick pair: A-U, U-A, C-G, or G-C.

[No sharp turns.] The ends of each pair are separated by at least 4intervening bases. If (bi, bj) ∈ S, then i < j - 4.

[Non-crossing.] If (bi, bj) and (bk, bl) are two pairs in S, then wecannot have i < k < j < l.

Free energy. Usual hypothesis is that an RNA molecule will form thesecondary structure with the optimum total free energy.

Goal. Given an RNA molecule B = b1b2…bn, find a secondary structure Sthat maximizes the number of base pairs.

approximate by number of base pairs

RNA Secondary Structure: Examples

Examples.

C

G G

C

A

G

U

U

U A

A U G U G G C C A U

G G

C

A

G

U

U A

A U G G G C A U

C

G G

C

A

U

G

U

U A

A G U U G G C C A U

sharp turn crossingok

G

G≤4

base pair

RNA Secondary Structure: Subproblems

First attempt. OPT(j) = maximum number of base pairs in a secondarystructure of the substring b1b2…bj.

Difficulty. Results in two sub-problems. Finding secondary structure in: b1b2…bt-1. Finding secondary structure in: bt+1bt+2…bn-1.

1 t n

match bt and bn

OPT(t-1)

need more sub-problems

Dynamic Programming Over Intervals

Notation. OPT(i, j) = maximum number of base pairs in a secondarystructure of the substring bibi+1…bj.

Case 1. If i ≥ j - 4.– OPT(i, j) = 0 by no-sharp turns condition.

Case 2. Base bj is not involved in a pair.– OPT(i, j) = OPT(i, j-1)

Case 3. Base bj pairs with bt for some i ≤ t < j - 4.– non-crossing constraint decouples resulting sub-problems– OPT(i, j) = 1 + maxt { OPT(i, t-1) + OPT(t+1, j-1) }

Remark. Same core idea in CKY algorithm to parse context-free grammars.

take max over t such that i ≤ t < j-4 andbt and bj are Watson-Crick complements

Bottom Up Dynamic Programming Over Intervals

Q. What order to solve the sub-problems?A. Do shortest intervals first.

Running time. O(n3).

RNA(b1,…,bn) { for k = 5, 6, …, n-1 for i = 1, 2, …, n-k j = i + k Compute M[i, j]

return M[1, n]}

using recurrence

0 0 0

0 0

02

3

4

1

i

6 7 8 9

j

0 0 0

0 0

02

3

4

1

i

6 7 8 9

j

CUCCGGUUGCAAUGUC n= 16((.(....).)..).. 0 0 0 0 0 1 1 1 1 1 2 2 2 3 3 3 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

E.g.:OPT(6,16) = 2:

GUUGCAAUGUC(.(...)...)

E.g.:OPT(1,6) = 1:

CUCCGG(....)